From d51159ad9eb6ad4c534c4b9500d1b296d4bd3810 Mon Sep 17 00:00:00 2001
From: Arthur DANJOU <arthurdanjou@outlook.fr>
Date: Mon, 3 Mar 2025 16:34:44 +0100
Subject: [PATCH] TP2 in Numerical Opti

---
 .../ComputerSession2.ipynb                    | 88 ++++++++++++++-----
 1 file changed, 67 insertions(+), 21 deletions(-)

diff --git a/M1/Numerical Optimisation/ComputerSession2.ipynb b/M1/Numerical Optimisation/ComputerSession2.ipynb
index 44dba70..17ee96a 100644
--- a/M1/Numerical Optimisation/ComputerSession2.ipynb	
+++ b/M1/Numerical Optimisation/ComputerSession2.ipynb	
@@ -288,44 +288,90 @@
   {
    "metadata": {
     "ExecuteTime": {
-     "end_time": "2025-02-18T17:42:16.515141Z",
-     "start_time": "2025-02-18T17:42:16.461572Z"
+     "end_time": "2025-02-25T21:29:40.198527Z",
+     "start_time": "2025-02-25T21:29:40.188272Z"
     }
    },
    "cell_type": "code",
    "source": [
-    "def total_time(x):\n",
-    "    # t = d / v\n",
-    "    run_speed = 8\n",
-    "    swim_speed = 3\n",
-    "    distance_run = x\n",
-    "    distance_swim = np.sqrt((6 - x) ** 2 + 2 ** 2)\n",
-    "    return distance_run / run_speed + distance_swim / swim_speed\n",
+    "import numpy as np\n",
     "\n",
     "\n",
-    "def d_total_time(x):\n",
-    "    run_speed = 8\n",
-    "    swim_speed = 3\n",
-    "    distance_swim = np.sqrt((6 - x) ** 2 + 2 ** 2)\n",
-    "    return 1 / run_speed - (6 - x) / (swim_speed * distance_swim)\n",
+    "# Objective function\n",
+    "def objective_function(x):\n",
+    "    return x / 8 + np.sqrt((6 - x) ** 2 + 4) / 3\n",
     "\n",
     "\n",
-    "x0 = 6.0\n",
-    "optimal_x, iterations = Newton(total_time, d_total_time, x0)\n",
-    "print(f\"Optimal distance to run: {optimal_x} km\")\n",
-    "print(f\"Number of iterations: {iterations}\")"
+    "# Gradient of the objective function\n",
+    "def gradient(x):\n",
+    "    return 1 / 8 + (6 - x) / (3 * np.sqrt((6 - x) ** 2 + 4))\n",
+    "\n",
+    "\n",
+    "# Hessian of the objective function\n",
+    "def hessian(x):\n",
+    "    return -((6 - x) ** 2 + 4) / (3 * ((6 - x) ** 2 + 4) ** (3 / 2))\n",
+    "\n",
+    "\n",
+    "# Newton's method for optimization\n",
+    "def newton_method(initial_guess, tolerance=1e-6, max_iterations=100):\n",
+    "    x = initial_guess\n",
+    "    iterations = 0\n",
+    "    while iterations < max_iterations:\n",
+    "        grad = gradient(x)\n",
+    "        hess = hessian(x)\n",
+    "        if np.abs(grad) < tolerance:\n",
+    "            break\n",
+    "        x = x - grad / hess\n",
+    "        iterations += 1\n",
+    "    return x, iterations\n",
+    "\n",
+    "\n",
+    "# Dichotomy method for optimization\n",
+    "def dichotomy_method(func, aL, aR, eps=1e-6, max_iterations=100):\n",
+    "    iterations = 0\n",
+    "    while (aR - aL) / 2 > eps and iterations < max_iterations:\n",
+    "        b = (aL + aR) / 2\n",
+    "        if func(aL) * func(b) < 0:\n",
+    "            aR = b\n",
+    "        else:\n",
+    "            aL = b\n",
+    "        iterations += 1\n",
+    "    return (aL + aR) / 2, iterations\n",
+    "\n",
+    "\n",
+    "# Initial guess for Newton's method\n",
+    "initial_guess_newton = 3\n",
+    "\n",
+    "# Run Newton's method\n",
+    "optimal_point_newton, iterations_newton = newton_method(initial_guess_newton)\n",
+    "print(f\"Optimal point (Newton): {optimal_point_newton}\")\n",
+    "print(f\"Objective function value at optimal point (Newton): {objective_function(optimal_point_newton)}\")\n",
+    "print(f\"Number of iterations (Newton): {iterations_newton}\")\n",
+    "\n",
+    "# Initial interval for dichotomy method\n",
+    "aL, aR = 0, 6\n",
+    "\n",
+    "# Run dichotomy method\n",
+    "optimal_point_dichotomy, iterations_dichotomy = dichotomy_method(objective_function, aL, aR)\n",
+    "print(f\"Optimal point (Dichotomy): {optimal_point_dichotomy}\")\n",
+    "print(f\"Objective function value at optimal point (Dichotomy): {objective_function(optimal_point_dichotomy)}\")\n",
+    "print(f\"Number of iterations (Dichotomy): {iterations_dichotomy}\")"
    ],
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Optimal distance to run: inf km\n",
-      "Number of iterations: Method diverges\n"
+      "Optimal point (Newton): 6.809045443967482\n",
+      "Objective function value at optimal point (Newton): 1.5702779015852644\n",
+      "Number of iterations (Newton): 7\n",
+      "Optimal point (Dichotomy): 5.999999284744263\n",
+      "Objective function value at optimal point (Dichotomy): 1.416666577259742\n",
+      "Number of iterations (Dichotomy): 22\n"
      ]
     }
    ],
-   "execution_count": 104
+   "execution_count": 14
   },
   {
    "metadata": {},